How to Derive and Apply Brewster Law in Physics Numericals
Brewster’s Law is a foundational concept in optics and polarization, explaining how and when light becomes completely polarized upon reflection at the boundary between two transparent media.
This topic is especially relevant for understanding natural phenomena and for applications in optical instruments and devices. When ordinary (unpolarised) light strikes a smooth surface like glass or water at a specific angle, the reflected light can become fully polarized.
This angle is called the Brewster angle or the angle of polarization, and it is closely connected to the refractive indices of the materials involved. The phenomenon was first formally described by the Scottish physicist Sir David Brewster.
At the Brewster angle, the reflected and refracted rays make a right angle (90°) with each other. As a result, the reflected light vibrates in a single plane, perpendicular to the plane of incidence, illustrating complete polarization by reflection.
Detailed Explanation and Physical Concept
- When light encounters the surface of a transparent material such as water or glass, part of the beam is reflected, and part is refracted into the second medium. Usually, the reflected light is only partially polarized.
- However, at one particular angle of incidence—the polarizing or Brewster angle—the reflected light becomes perfectly plane-polarized.
At this special angle, the direction of the refracted ray is perpendicular (90°) to the direction of the reflected ray. - The vibrations (electric field) of the reflected light are entirely perpendicular to the plane containing the reflected and refracted rays. This specific behavior of light is captured mathematically by Brewster’s Law.
The relationship helps explain how devices like polarizing sunglasses block glare and why certain surfaces shine more strongly when viewed at particular angles.
Brewster’s Law: Formal Statement and Formula
Brewster’s Law states that the tangent of the angle of polarization (p) is equal to the ratio of the refractive indices of the two media. If ordinary light of a single wavelength travels from one transparent substance (with refractive index n1) into another (with refractive index n2), the law can be written as:
tan p = n2 / n1
Here,
- p = Brewster angle (angle of polarization)
- n1 = refractive index of first medium (e.g., air)
- n2 = refractive index of second medium (e.g., glass or water)
Step-by-Step: How to Solve Problems Using Brewster’s Law
To use Brewster’s Law in calculations or conceptual questions, follow these steps:
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Identify refractive indices: Note down n1 and n2 for the two media (for example, n1 = 1 for air, n2 = 1.5 for glass).
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Write the Brewster’s formula: tan p = n2 / n1.
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Calculate the Brewster angle: Take the inverse tangent (arctan) of the ratio to find p.
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State the orientation: At this angle, reflected and refracted rays are always at 90° to each other.
Application Table: Calculating Brewster Angle for Various Materials
| Medium 1 (n1) |
Medium 2 (n2) |
Formula | Brewster Angle p |
|---|---|---|---|
| Air (1.00) | Glass (1.50) | tan p = 1.50 / 1.00 | p = arctan(1.5) ≈ 56.3° |
| Air (1.00) | Water (1.33) | tan p = 1.33 / 1.00 | p = arctan(1.33) ≈ 53° |
| Air (1.00) | Diamond (2.42) | tan p = 2.42 / 1.00 | p = arctan(2.42) ≈ 67.5° |
Sample Problem: Worked Example
Question: Light is incident on a glass surface (n = 1.5) from air (n = 1.0). Calculate the angle of incidence at which the reflected light is completely polarized.
Solution: Using Brewster’s Law:
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tan p = nglass / nair = 1.5 / 1.0 = 1.5
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p = arctan(1.5) ≈ 56.3°
Therefore, at an incident angle of about 56°, the reflected ray from glass is fully polarized.
Visualizing Brewster’s Law: Orientation of Rays
| At Brewster Angle | Physical Outcome |
|---|---|
| Reflected and refracted rays | Make 90° (are perpendicular to each other) |
| Reflected ray | Fully polarized perpendicular to plane of incidence |
| Practical result | Glare is minimized, commonly used in polarizing sunglasses and photography |
Key Points and Applications
- Brewster’s angle is also known as the polarizing angle.
- The law is crucial in mastering concepts of Polarisation of Light and Optics.
- Polarizing filters and anti-glare devices exploit this law.
- It is commonly tested in competitive exams and forms the basis of several practical and theoretical optics problems.
Important Related Formulas
| Concept | Formula | Use |
|---|---|---|
| Brewster’s Law | tan p = n2 / n1 | Find angle for complete polarization |
| Snell’s Law | n1 sin i = n2 sin r | Relate angles and refractive indices at a boundary |
Recommended Resources and Next Steps
- Read more about Polarisation of Light and Light Reflection and Refraction for foundational understanding.
- Practice numerical problems and concept questions using this Brewster Law resource and related optics pages.
- Explore Vedantu’s Optics lessons for deeper conceptual learning and exam preparation.
Summary
Brewster’s Law clearly explains the condition for achieving complete polarization by reflection. By applying the formula tan p = n2 / n1, you can easily determine the unique angle where this phenomenon occurs for any two transparent materials. Understanding Brewster’s Law not only deepens your knowledge of polarization but also builds your problem-solving skills in optics—a critical part of modern physics and real-life applications.
FAQs on Brewster Law Explained: Formula, Proof & Exam Questions
1. What is Brewster’s Law?
Brewster’s Law states that when unpolarized light strikes a transparent surface at a specific angle (called the Brewster angle), the reflected light is completely polarized perpendicular to the plane of incidence. This phenomenon occurs when the angle between the reflected and refracted rays is 90°. The law is expressed as: tan θB = n2 / n1, where n1 and n2 are refractive indices of the two media involved.
2. What is the Brewster angle and how is it calculated?
The Brewster angle (also called polarizing angle) is the angle of incidence at which reflected light becomes perfectly polarized.
Calculation:
tan θB = n2 / n1,
where θB is the Brewster angle, n1 is refractive index of the first medium, and n2 is the refractive index of the second medium.
Use the inverse tangent function to find θB after substituting values.
3. State and prove Brewster’s Law.
Statement: Brewster’s Law states that the tangent of the angle of incidence at which light reflected from a transparent medium is perfectly polarized is equal to the ratio of the refractive indices of the two media.
Proof:
- At the Brewster angle, the reflected and refracted rays are perpendicular: θr + θt = 90°.
- Applying Snell’s Law: n1sinθB = n2sinθt.
- Since sinθt = cosθB, we get: n1sinθB = n2cosθB.
- Dividing both sides by cosθB: n1tanθB = n2.
- Therefore, tan θB = n2 / n1.
4. What is the difference between Brewster’s Law and Snell’s Law?
Brewster’s Law deals with the polarization of reflected light and gives the condition for maximum polarization, while Snell’s Law relates the angles of incidence and refraction regardless of polarization.
Main differences:
- Brewster’s Law: tan θB = n2 / n1; used for finding the angle of maximum polarization.
- Snell’s Law: n1sinθ1 = n2sinθ2; used for general refraction at an interface.
5. How are reflected and refracted rays oriented at the Brewster angle?
At the Brewster angle, the reflected and refracted rays are oriented at right angles (90°) to each other. This means the angle between the reflected ray and the refracted ray is exactly 90°, which leads to maximum polarization of the reflected light.
6. Where is Brewster’s Law used in real life?
Brewster’s Law is applied in several real-life contexts:
- Anti-glare sunglasses – to block polarized light and reduce glare
- Photography – using polarizing filters to enhance image contrast
- Optical instrument design – to control unwanted reflections
- Laser technology – employing Brewster windows to ensure polarization
7. What is polarization by reflection?
Polarization by reflection refers to the process where light, after reflecting off a non-metallic surface at the Brewster angle, becomes polarized perpendicular to the plane of incidence.
- At this specific angle, the reflected beam contains vibrations mainly in one direction.
- This principle is used in sunglasses and camera filters.
8. How do you solve numerical problems using Brewster’s Law?
To solve Brewster’s Law numericals:
- Identify refractive indices n1 and n2 of both media.
- Apply the formula: tan θB = n2 / n1.
- Calculate θB using the inverse tangent: θB = arctan(n2/n1).
- Substitute correct values and compute the answer as per the syllabus.
9. What are some typical exam questions on Brewster’s Law?
Common exam questions include:
- Numerical calculation of Brewster angle for given media
- Derivation or proof of Brewster’s Law using Snell’s Law
- Describing the orientation of polarized light at the Brewster angle
- Conceptual difference between Brewster’s and Snell’s Laws
- Application-based problems on polarization in optics
10. What does the Brewster’s Law formula represent physically?
The formula tan θB = n2/n1 defines the angle of incidence (Brewster angle) at which light reflected from a transparent surface is fully polarized perpendicular to the plane of incidence.
- At this angle, maximum polarization occurs in the reflected ray.
- It is widely used in optics, lasers, and polarization devices.
11. Can Brewster’s Law be applied to all types of surfaces?
Brewster’s Law applies mainly to smooth, non-metallic, transparent surfaces like glass and water.
- It is generally not applicable to metallic surfaces, as they do not produce perfectly polarized reflected light due to free electrons causing absorption and other effects.
12. Why is understanding Brewster’s Law important for Physics exams?
Brewster’s Law is a frequently tested concept in Physics exams such as JEE, NEET, and CBSE boards because:
- It connects optics, polarization, and law derivations as per the syllabus.
- Mastery improves both conceptual and numerical problem-solving.
- Questions on this topic are high-weightage and often require detailed explanations and calculations.
